Tensor products of maximal degenerate series representations of the group SL(n, R)

Projective pseudodifferential analysis and harmonic analysis

“…Equation (1.49) shows that generalized eigenvalues of the operator in the title of this section present themselves in the form −(ρ + n 2 ) 2 : more to the point, the spectral theory of this operator-as developed in [1,2] with the help of H n -spherical distribution theory-shows that the spectrum of Δ n has a continuous part, consisting of all numbers ρ(−n − ρ) with ρ = − n 2 + ir ∈ − n 2 + iR, and a discrete part consisting of the numbers…”

“…As shown in [1,2], the quasiregular representation of G n in L 2 (X • n ) = L 2 (G n /H n ) decomposes into a continuous part and a discrete part, a fact tantamount to the analogous statement regarding Δ n . We shall take it, temporarily, for granted that-a consequence of the analysis to be developed in the next section-functions on X • n in the continuous part of the decomposition can always be viewed (in many ways) as restrictions to X • n of R × -invariant functions satisfying in R n+1 × R n+1 , in the distribution sense, the equation…”

Covariant quantization: spectral analysis versus deformation theory

“…Berezin, and was highlighted in a series of papers by van Dijk and Hille [34], van Dijk and Molchanov [36,37] and Unterberger and Upmeier [162] as well as in [38].…”

“…Such a quantization was studied-in a way independent from the superselection method-in [161] in the case when n = 1 and, in the general case, in a series of papers [36,37]. In all cases, the development had to include a description of the decomposition of L 2 (G n /H n ) under the quasi-regular representation of G n , which amounts to a description of the spectral decomposition of the basic G n -invariant differential operator Δ n on • n .…”

The exterior and symmetric square of the reflection representation of An(q) and Dn(Kable1, Sanat2 2005Journal of Algebra4000View full textAdd to dashboardCite

“…Within these families the general ClebschGordan problem has been solved only for groups of small rank [1][2][3]24,25,27,29,30]. For unrestricted rank, attention has typically focused on tensor products of small representations, both because they tend to possess interesting structural properties, such as being largely multiplicity free, and because they arise in applications [10,16,31,35,38]. The little evidence that is available for large representations of groups of unrestricted rank suggests that their tensor products are relatively insensitive to the factors and exhibit little fine structure [36].…”